David Gross: Millennium Prize Problem: Yang Mills Theory

Well, welcome to the grand finale, the final conference in the series on the

problem

s of the

millennium

prize

and we are very grateful that kitv4 is hosting this event on KITV. Our speaker today really needs no introduction. It is a great honor and pleasure to have Nobel laureate, former director, professor of KTP, David. Gross, who is going to tell us his perspective on the

millennium

problem

? Yaya's

theory

, well, when they asked me to do this, it was a year ago or something, it was one of these invitations until now in the future, you would say sure and then a few days ago I started thinking what the hell am I going to tell them? tell these mathematicians about quantum field

theory

, so I wrote a talk and then tore it up.

Another one was last night and since it's a math talk I'll use a whiteboard. and since it's KTP, if there are any questions, stop me and ask me, so I assume you already solved the other six problems and today's topic is probably the most difficult or at least that's what Edward Witten tells me about the seven millennia. Awards now Arthur Jaffe was an old friend and colleague of mine when he started putting together the clay base before he had a brilliant idea of ​​how to get a lot of publicity, as you know, for the Clay Institute.

I think it's called formuland. awards that would be announced at the beginning of the millennium, then you would know the clay base, it gets a lot of publicity and it sparked a lot of interest in mathematics and maybe even these problems, but you don't have to pay money and so far They have been very successful, they have gotten millions of dollars in advertising and they've just had to pay a price that someone they maybe didn't even know wouldn't accept, so it hasn't cost them anything, but it's clearly generated a lot of interest and to win the awards remarkably, I mean, some of us We physicists were surprised that two of the seven

prize

s were in physics, one of them, as you know, the seven Navier-Stokes equations.

I put together seven four different formulations of that award and I certainly haven't. It is still unsolved and then a prize for the proof of the existence and a mass limit in a non-Abelian Yang-

mills

theory or which is not a surprise given that Arthur Jaffe was one of Clay's presidents and was in that moment and Edward On his board, this is a prize, a problem that is even difficult to formulate. I mean, you could point out what they said, which I don't think I even have, it's more or less to prove the existence of a known theory.

I've committed to the theory and the four dimensions for an arbitrary age group and to prove the existence of a mass gap, but to even explain what that prize is takes more than one lecture and probably an entire course, and I'm going to give a very limited lesson. really explaining even what the problem is cannot be done adequately in an hour or a semester, but I will try to give a certain perspective on it, but I would like to say a few words only from the point of view of a Physicist who was educated in a period where mathematics was many of these ideas that form the basis of this prize were being developed, but the attitude toward mathematics was quite different, one of my mentors, Murph Goldberger at Princeton, wrote in his famous book on dispersion.

Relations that were an alternative and even more opaque approach to the problems are now adjusted by quantum field theory. He wrote that mathematics is an interesting intellectual sport, but it should not be allowed to get in the way of obtaining sensitive information about physical processes, which in general is the attic of what or at least used to be there for most physicists and us. allowed you to go very far on fairly flimsy foundations, the foundations are still quite flimsy and that is because the essence of this problem that the problem is asking allows you to actually have the formulation somewhere has included it in the abstract, but speaking in terms general, it is the existence in the sense of quantum field theory of Mills theory in four dimensions or dimensions with a gauge group G, an arbitrary semisimple in each group g plus the finite mass limit now get rid of the theories are examples of quantum field theories, the ones we use to describe the nature of aya, the standard model and four dimensions have never been shown to exist, in fact I'm not even sure they are non-trivial, except in very special cases of enormous Se has shown that there are symmetry quantum field theories of any kind, so this is not a problem you know, it is a more complicated problem where there are examples of what an existence proof would consist of and attempts in the past.

I must say that it coincided with another quote from my mentor Goldberger where he discussed the proofs of dispersion theory which, according to him, are like man's nipples, neither useful nor profitable, and in general, that has always been my attitude towards the proof. of the existence of quantum field theory if in fact, if you look at the literature on what has been shown for quite trivial quantum field theories, we certainly don't have the richness of this that allows the second step, they are ugly and useless and, Well, I must admit that many of my colleagues do not.

You're not that excited about this clay prize, yeah, no, now because you're not, you're not really sure what you're talking about, other than what existence is kind of okay if you don't worry about all those details. which we will consider is something essential to describe the real world and our then, and it is easy, so it is trivial, yes, I mean, you know that mathematics is not beautiful or useful, that is, none of the techniques, that is , many people have done an incredible job. There is a huge amount of work to try to prove existence and I am not going to say too much about this as I find it very urgent, but I will say a little and the reason this award has been formulated is that I think it is of great interest .

After I leave you, I'll say why it's of interest: One of these theories we're talking about is twofold: Do you know the theory? The standard model of elementary particle theory is well tested and, in some sense, is the best physical theory. The most complete, ambitious and successful theory we have ever had. We would like to know if it is based on solid foundations. That's one, but we kind of know it is. I mean, physicists have ways of discovering the truth, and don't they, like I said. in my first quote, which is not guided solely by mathematical rigor, if in the first part of this problem it is difficult to know how it would fail, since it is difficult to know what it would mean to prove the non-existence of quantum field theory, it could not succeed . in proving it as people have done for many years, yes and that is indeed one of the problems and I will say a few words about that, but not in great detail because it involves too much background, but it would be possible to prove the existence of for example, the continuous limit of a well-defined way of defining

yang

-

mills

theory and and and to show that it does not have a finite mass scale, so it would be possible to arrive at a negative result here for the combined problem and the Of course, we would be surprised to everyone, since we know it is false, but our evidence is based on a way of thinking that many mathematicians find difficult to assimilate and when I started to think about how to give a lecture like this, I went back and looked at the attempt that Witten and others at the Institute did an experiment ten years ago at Princeton to try to educate and try to teach a group of mathematicians quantum field theory and string theory eighteen years ago, in fact, just before I left Princeton, I even said a conference in that single standards group. and students like David Cosgrove who took my class rewrote the lecture notes and I couldn't read them, I understand that afterwards it's impossible, more or less, anyway, it wasn't a great success because you know we think in a way that is based in, you know, something like that. walking on a rope over a cliff without looking down and I get pretty far that way because of the success story of physics, the fact that we have nature to guide us, so we know these things are correct with certainty based on to a lot of internal knowledge. analog examples of consistency in other simpler cases of physics and an experiment, okay, that's why it could still be interesting, so if one found a profile counter in the mass gap here it would be a total shock and you know , I will make strong bets against it.

It's a million dollars, but I think what's most interesting is the sense that if mathematics could be developed to answer such questions, given the beauty of the theory, physicists consider those methods to be powerful enough to answer these questions. , but they have to be beautiful at the same time. and useful, but it's clear that no one has really good ideas, as far as I could see, or mathematical tools, so the prize could be useful in pushing mathematicians to develop those tools and they will probably be very different than anything that has been proven so far, so I have to tell you a little bit about gauge theories and the gauge theory that I'm sure you know is electromagnetism, which was the first field theory in which there were dynamical objects that were functions if The lack of space-time could be measured and could influence other objects and Maxwell's theory of electrical magnetism is based on what we call a gauge theory for historical reasons.

Not Abelian. An abelian gauge theory based on the u1 gauge group. Its dynamic field is a connection. in a u1 packet over Minkowski space, so space and time is a four-dimensional three-dimensional space with Lorentzian signature and the physical observables are field strengths, so this is a unique form and the intensity of field that is only given is a form whose components are the familiar electrical ones. fields and magnetic fields, so Maxwell unified electrical magnetism in this theory whose equations of motion are simple by definition PDE of second order that are derived from a principle of action of which is only the integral in the space of this curvature of this connection squared or now Maxwell's field equations were the first field theory invented and that concept of having dynamics described by a function of space and time like a magnetic or electric field has dominated physics since it was also the first theory .

Its equations of motion and action are invariant under Lorentz transformations. one of the eyes, AMA tree of this plane, this Minkowski space with plane metric, these linear equations have wave solutions which, as Maxwell discovered, are just light, the waves of waves and this field strength is governed by the equations Maxwell's equations, so in the absence of sources, Maxwell's equations are loaded in a vacuum is very simple, easily soluble, is described as non-linear waves that pass through each other, do not interact and travel at the speed of light and all the theories that we will discuss as we listen to those theories are generalizations of that, so this is a dynamic variable, the strength of the connecting field.

In this action there is a constant that plays no role classically since it does not affect the equations of motion, but it is known as electric charge and if we introduce objects that transform non-trivially in this age group carry electric charge, they will be related. to the electric charge of that field, but we are going to discuss it again in the solution and classically this charge is irrelevant now at the classical level the theory is very simple linear equations very symmetric actually has a larger symmetry group than the Lorentz transformations due to the lack of an explicit length scale here, so since we're going to talk about masses and couplings, let's say a few words about physical units, you should always specify the units of the physical objects you're talking about so that the objects that can be measured, you measure things, you measure, you compare a meter, you measure length, you measure time, some counts of some clock beats and you measure energy or mass, so, MKS, meter, kilogram, second, CGS, which is what you need in physics, always to specify the units we do. of course, with amorphous units of response, but in fundamental physics we tend to use units in which at least two of the basic dimensional constants of nature appear, one is the speed of light and the other is the Planck back, which has units of mass times length squared divided by time and we can define units in which these are 1 with respect to these as units of time and getting rid of two of these fundamental units express everything all velocities are fractions of the velocity of the light which is 1 in these units all actions mass times velocity Lens times are also expressed in units of Planck's constant, which plays a fundamental role in quantum mechanics.

Sometimes we complete this with another unit, but we won't hear that the natural unit is Newton's constant, which defines the unit of mass. We will not talk about severity. Not at all, using units like this means that everything can be expressed as a length. Well, time is a length divided by the speed of light, so now it's easy to see that the correct dimension for thegauge field is one over the length and for the electric. field or the magnetic field that has an additional derivative one over the length squared and therefore you can see that F has dimensions of one over the length and the fourth volume integral has dimensions of length up to the fourth and so Therefore, this charge parameter here has dimensions in these units the way we normally say is that the electric charge squared on the bar H (see Planck's constant divided by the speed of light) is dimensionless, so e squared has dimensions of the bar h, but setting them to one and is as common a number as physically defined 101 over one hundred. thirty seven point zero something zero three five then it is a small number and it has no dimensions and as you see in this theory there are no parameters that have dimensions of length under a scale of all lengths the volume increases as the scale factor to the fourth power. time scales like length etc., this scale is invariant under rescaling of all length, all times, all energies and E is dimensionless, it is just a pure number in these units, so this theory is scale and in fact it is rather invariant as the complete symmetry of this action is like this4 point 2 which is organizing transformations plus position scaling plus special conformal transformations inversions so that is for electrodynamics in a vacuum, which is very trivial, it is even trivial to quantize this theory , this is essentially a theory of non-interacting infinite, the new moving number of non-interacting harmonic oscillators and before a transformation, one cannot easily construct a Hilbert space that describes the quantum mechanics of long in M ​​in the vacuum. now and this was the first quantum field theory that was attacked after the birth of quantum mechanics, electrodynamics existed very successfully in classical times, but obviously it had to be quantized and in a sense it was already quantized when Einstein applied Planck's ideas to radiation so coupled to electrons that They were also described by fields.

I'm not going to describe matter here, so I'm not going to go into fermionic fields and describe the electrons and the rock equation, but coupled to matter, electrodynamics was no longer so linear due to the coupling of the effects of mechanics. quantum. pictorially we say that two rays of light described here by these wavy lines or quantum particles description of these fields can interact creating pairs called virtual pairs in the vacuum of charged particles such as electrons that then Reece quatre and these effects were a challenge to In reality, it is It takes something like 30 years to really control and calculate for many technical reasons, but in the case of electrodynamics, this was developed quite successfully and has led to the most extraordinary tests of fundamental physics ever calculated, e.g. the electron that has rotates 1/2 so it is a intrinsic rotation also due to its charge as an intrinsic magnetic moment that is measured in some dimensional units so it is called G and that for a relativistic electron that does not interact with the field electromagnetic is one and there was a deviation the measured deviations from this from a which led to the need to develop QED as a quantum theory and is now measured with a 12-digit point, so this is the actual measured value of the intrinsic magnetic moment of the electron 1 point zero zero one one five nine six five two one eight zero eight five and the last two digits are uncertain plus minus seventy six this is the experimental value it is quite incredible that an experimenter can make a measurement of this small magnet at one part in a billion not only does it not really need to be written down the theoretical value, which again is a tour de force of calculation because it agrees to this extent with a measured value.

In fact, in theory it can't be tested with this level of precision in experiment because we don't know. the fine structure constant the parameter that goes into this theory with enough precision in fact you can turn it around use the experiment to get one of the best determinations of the fine structure constant now this is the case although no one has proven its existence of QED as a quantum version of electromagnetism couples to electrons that we don't believe really exist and to this and ultimately this is the result of an expansion in the powers of this fine constant structure with coefficients we call them GN, which we believe, although again no one has demonstrated.

Go as n factorial for large n and therefore this series that was being added to get this number as zero convergence radius, so here's a good example of why mathematicians and physicists find it difficult to communicate that we are talking about QED, which I haven't really done. described because I have not told you how to couple these E and B gauge fields to matter does not really exist, we believe that for good physical reasons, the same reasons that we believe that the Yang-mills theory is not Abelian, this theory is exists, furthermore, even if it existed, we don't really know how to calculate well, the only control we really have is this perturbative expansion in powers of this very small number, fortunately small and we have good reasons to believe in physical reasons, as well as analogies with other quantum field theories. with the same structure they definitely have this divergent behavior that these perturbative expansions we use to one part in a billion radius of convergence 0 which we don't know we don't know either but anyway I'm not going to complicate either n or n factorial n over 2 factorial , so this is, in a sense, the most spectacular example of the ability of physics to predict quantitatively make predictions that can be tested and agreed that it is a basic test, but it is an example where we don't actually we believe that the theory makes sense without further input that I haven't told you about or or or the calculation method is under control now, yes, well, let me, let me, let me, just, yes, well, you know, the answer is.

So the reason we see this today is that when we say here the existence of the theory you know, what is called for in the clay prize is something that most quantum field theories, most Visitors would say today that it is too much to Ask, what we are really interested in is not quantum field theory, what effective field theory, which means that quantum field theory assumes the existence of fields functions of points in space-time and here, in the field formulation, this actually depends on space. -now also this here we are taking products of fields at the same point now it turns out that said products are extremely singular was one of the technical difficulties of the theory but the fact that we have products of fields at the same point Which means that we are really taking these points seriously.

We're really saying that we believe physics is formulated on a continuum where we can bring fields of objects arbitrarily close together. Now we have no evidence of that because at best we can look at a few centimeters. With our eyes, microscopes, accelerators, we can travel distances from 10 to minus 18 centimeters, but there's a long way between 10, minus 18 and 0, especially on a logarithmic scale, and it's a bit arrogant to say you should imagine that. This is described by a quantum field theory which assumes that it makes sense to talk about these physical objects up to arbitrarily small distances - sorry, we don't really, but anyway, the modern view would be to say, let's be more humble. and you know, we put a limit not on zero but on the smallest distance that we've probed, we haven't seen anything that destroys the point structure of space-time, but there could be, and then that eliminates a lot of problems and allows us to define a theory without destroying it if we want the continuum at very short distances, for example, replacing the continuum with a set of discrete points, as we will do shortly to find Mills' theory with the network.

What we've learned is that if the limit, say, is 10 to the power of minus 33 centimeters to choose an arbitrary distance, it's not going to affect the physics much here at these much larger distances and the non-existence of QED that we'll get to later occurs. at distances that turned out to be, say, on a distance scale cut from e to - 1 over alpha times some standard distance scale, let's say that if you want the radius of the electron distance scale to be defined by the size of the electron, it is 10 to the minus 11 centimeter, so this is, you know, e to the minus 137, an extraordinarily short distance, much smaller than anything that happens here, this is where the problems we run into will kill us, we think that it won't show you that this QED, this simple Abelian gauge theory, exists as a well-defined theory and, furthermore, it's fine, so we don't have to worry about the theory existing.

A much less ambitious theory, one that cuts on a credibly short scale. Are there no regiments? We do. There is a much larger scale and we can. argue that it is an asymptotic expansion, it is perfectly valid for measurements made on large scales and that, although it is not convergent, it is asymptotic, so it is an asymptotic expansion in which if these coefficients a and grow as factorial n until they reach the 2n of order 1 over alpha this should give increasingly better results as asymptotic expansions tend to give and the inherent errors we make will be from order e to alpha well to order one over alpha so until we get to this it is to obtain this result you have to work in fourth or fourth or fifth order fourth order I think, but for loops and until you get to the order you know maybe a hundred loops you're going to have a problem, no one is going to do that so you can use a synthetic expansions in which you don't have any rigorous control over the errors and you can't make the precision arbitrarily good and, furthermore, the diseases that prevent this theory from really existing occur only at some incredibly high limits and we know for many reasons that there will be other physical effects, other phenomena that will change things in these high energies.

Now, the Yang-Mills theory I'm referring to now we're going to demand a lot more of one. Actually, for this, for the same reasons as us. believing that electromagnetism has a complete quantum field theory carried to arbitrarily short distances does not exist, we believe, and that is the first part of the proof of Abelian gauge theories, even non-Abelian gauge theories of the vacuum which are highly non-trivial. exist and you can actually go to arbitrarily short distances it's a complete theory, you don't have to think of it as just a long distance approximation to something you don't know so you cut it out and secondly it has this remarkable factor of producing a scale which this theory in vacuum does not have dynamically in quantum mechanics, so we are going to discuss the non-Abelian version of electromagnetism, it is as if electromagnetism is wickedly scaleless like a dimensionless coupling and yet it will produce a scale with a limit of mass effects of quantum effects Now, what does it really mean to build, define, or prove the existence of a quantum field theory in a way that no one has ever done except in a very trivial example or in examples where the symmetries are so strong that certain parts The quantum field theory can be built explicitly using the large symmetry, the inter kabila T.

It is difficult to explain this in great detail, but there are certain features that are necessary for any quantum theory. Open space states in Hilbert space are needed. You want the Hilbert space to have certain positive metric positive gauge states because our interpretation of quantum mechanics requires that you know that you identify the probability with the gauge states and the probability should be positive, it's trivial, it's not fun. and that is correct, then it is a theory, in fact, there have been theories in less than four dimensions like life or theory that, from our point of view, do not have these dimensionless couplings, they have dimensional couplings like masses or nonlinear couplings with mass dimensions and are simple enough over short distances that one can actually prove the existence of the theory now, as I said, they don't teach you anything interesting about the structure of quantum field theories.

We need to describe nature, but you know that there are very difficult jobs and you know that they required a lot. However, it doesn't seem like that effort allows you to tackle these more difficult methods, more difficult theories and I will say what has been done for four-dimensional theories, but not much anyway, the most important thing is that we want relativistic invariance, so they have a symmetry very similar to classical Lorentz invariance inm, in fact, the matrix and punk variance, which are translations of space and time plus Ren transformations, so we are not dealing with fermions like that3 and in quantum mechanics such symmetries of our theory are represented by operators so we want operatorsunitaries of Opera that under what states and the operators the fields that are transformed covariantly and we want a vacuum a state that is invariant under Lorentz transformations and translations the fundamental state of the system must be symmetric these are properties that I am naively certain and true using our limited tools to solve quantum field theories, but it should be true for what we are looking for.

We also want the generators of these symmetry transformations that are for translations, the energy of the moments and the moments to have defined. such particular spectral properties the energy that is a generator that the Hermitian operator should have and this Hilbert space should have a spectrum that is limited from under the vacuum is the invariance of the vacuum zero energy and zero momentum its translation in time under time and invariant spatial translations and in fact, the representation theory of the Lorentz group tells us that the states that transformed irreducibly under the Lorentz group are characterized by moments and other discrete labels and Kasmir, the energy squared minus the momentum squared is what we now call the mass of the particle if we look at In electromagnetism, this theory here we have an unbroken conformal truth of scale invariance under scale transformations.

Scalar length up energy scale down, so in the case of enm the energy spectrum is continuous, but that is not what we observe if we ignore the existence of light. rays photons we see particles of definite mass particles mass is just the energy of a particle momentum is zero, okay and the spectrum of the energy or a theory of which we are going to discuss in particular the spectrum of the mass operator as discrete set the vacuum with zero mass, zero energy, zero momentum and then the lowest particle mass, a discrete state with some mg/g representing space and then surely pairs of these particles with twice the mass and some energy additional kinetics, a continuum of multiparticle states. so in a quantum field theory where we have massive particles and not massless particles like light rays, there is a gap between the vacuum, the ground state and the first state that can be created with a minimum energy equal to the mass of that particle and then multi-particle states, perhaps other isolated discrete states. particle states as well and it is this gap that is the really interesting dynamical question in the second part of this problem, how to produce

yang

-mills theories that will classically have approximately the same structure as electromagnetism and classically have scale invariance and are controlled. to produce a mass gap in the power operator spectrum, but to be exotic anyway, why don't you feel slimmer?

Double-spaced, you would like to have states that are representations of the transformation in a coherent way under large transformations. I would like to have some kind of space and time dependent field operators that transform coherently under Lorentz transformations. You'd like to be able to define matrix elements of the product of these operators, etc., but that's a lot of trouble, sorry, well, the fact that you're here, you have a, so you're done, let me turn to the theory for a moment. , well, let's take one of these theories that is representative of the class of theories that fall into hyper QCD, it is an example of it, sorry. goes to zero like what goes to zero I'm sorry, the gap is the so, in any quantum field theory where you have, you know this configuration there is a gap, I mean, there is a well defined lower energy state and the spectrum of P is zero above orthogonal to the vacuum and that could be zero, it is in quantum electrodynamics, quantum electrodynamics, one has the vacuum and then a single ray of light can have an energy equal to its frequency and it can have a frequency as low as you wish, an arbitrarily long wavelength, so the state of a photon has any energy starting at zero to infinity.

The two-photon state has any energy starting at zero, you just continue, which is what you would expect because, after all, there is no scaling in the theory, no periodic limits. conditions and oh, you mean sorry if you put the theory in a box, well, we are not in a box for an hour three one now, when the yang-mills theory, so briefly, the yang-mills theory is exactly the same action except now we have a this is a connection now in some package some gauge group and su3 is the real world relevant gauge group for QCD but this problem arises for any gauge group but now I'm not a group of abelian gauges and field strengths. is nonlinear and the gauge field and is transformed covariantly under each transformation and the action involves plotting a quadratic form in the lie algebra of the same beast, but which is now like quadratic terms and cubic terms and quartic terms in the intensity of the field in the equations. of motion that are now d f or d f+ are not linear, but again, naively, this coupling is dimensionless and the classical theory is invariant not only under the need for large transformations but under conformal transformations and scale invariance and, therefore, classically, although it is very difficult to know the equations.

The classical angle equations are almost as complicated as the equations of design science, although somewhat simpler still, it is clear that the spectrum is continuous. Transformations under scale. The energy scales and theory are invariant now, when Yang-Mills theories were first proposed in the 1950s as a generalization of electromagnetism. Because people had discovered approximate symmetries, non-Abelian symmetries of nuclear particles, one of the reasons it was so difficult to imagine applying them in the real world was the fact that there were no particles with a continuous spectrum like this, except the photon. The only long-range arbitrary light particles that exist to obtain the origin of the long-range force are believed to be photons, light rays, or gravitons that were ignored by the quanta of Einstein's theory.

If we look at this classical theory, it would be a continuum. In the spectrum there would be some type of non-linear waves that would be like electromagnetism, except that they interact there. Two waves cannot be superimposed to obtain another solution in those equations. It's a terribly nonlinear theory, but only because symmetry clearly includes scaling of energies. There is a continuous spectrum and quantum mechanics that gives rise to the long-range interactions and forces that one was trying to describe, that is, the strong and weak nuclear forces that act within the nucleus, act within the nucleus in their short range. . and if they are to be described by quantum fields and it was not obvious back then, the quanta of those fields, the spectrum of the energy Hamiltonian, would have to have a gap, in fact, the value of this gap this mass is related through of the force range one over the force range mediated by such by particles of this mass, then this was the problem that we are trying to apply gauge theories to the real world was now solved in two different ways in the case of the weak interaction through the so-called Higgs mechanism that everyone has heard of, its final form has been verified.

In the case of the weak interaction, new degrees of freedom are added, another field, the Higgs field, is organized. so classically your minimum energy state breaks the underlying symmetry here and in that vacuum state these broken scales result and very to introduce the scale for sure, but you also create a gap classically, very simply, it's a mechanism classic that does not require a quantum. Mechanic and indeed most of us, I must say, 40 years ago were a little suspicious that such a cheap solution could explain the weak nuclear force, but it turned out to be the cheapest classical solution.

Survive quantum mechanics. Explain the weak nuclear force. called the Higgs mechanism, but what you have done is you know that you have added something else, but new parameters, you have broken this conformal symmetry scale by hand and generated a mass limit and then you can treat the theory in a very similar way to the calculation of the electromagnetism perturbatively with Virgin expansions, correlative corrections and they work, the theory exists well, it is probably not mixed with electromagnetism, it is actually a gauge theory su 2 cross u 1 which is diagonal u 1 and that is electromagnetism and as such , along with this Higgs field probably not.

It doesn't exist strictly either, but no one cares because we all know that at some point the physics is going to change and we can use that approximate theory way down the color, but then there are the strong interactions and that is a purely silent explanation of why the short-range forces and why the basic particles, the basic quanta of this field and the quarks I have not discussed, but these light rays of the three gauge group are not observable, which is another property of the theory about which I want this gas. called confinement which is a consequence of a rather surprising and beautiful quantum structure of the UCD and is also the reason why we believe that this, among all these other examples, is a theory that actually exists without any limits at arbitrarily short distances.

There is no physics that comes from these other forces and ultimately gravity, so who needs a mathematically rigorous, complete and exact theory? Anyway, it's nice to have one, what mathematicians, that's why they should prove it exists, so let's look at QCD a little. QCD is classically described by this connection in a su 3 package with a non-Abelian field strength and classically nonlinear equations. Now, classically, Mills' theory hasn't been studied a bit, but it's complicated, it's not linear, and it's not observable either. You know, classical electromagnetism is a thing. we use it all the time in the real world and that's actually because there is no ASCAP, so it's easy to excite photons, lots of photons and quantum.

The quantum mechanics of light quanta is well described by classical fields when the number of photons is very large and you can easily construct states with many, in fact an infinite number of photons without requiring an infinite amount of energy, but that is not It is true once there is a gap then you need a minimum amount of energy to build a quantum and a very large frequency. Remember the energy is like a frequency and if the minimum energy is approximately a minimum energy of what would be the lightest state of a QCD in the QCD vacuum, that minimum frequency would be something like 10 to the power of minus 24 Hertz, now it is quite difficult to shake something and create a classical chromodynamic field, I mean we don't make it unimaginable that there is some regime, a microscopic phenomenon that is described by classical QCD just because of this gap, okay, anyway, don't let me describe more which demonstrates the existence of a theory.

It won't let me write the answer, we know the answer to the theory as we use it. One of the reasons we believe in the positive answer to this problem is that this theory works in two ways. One of the predictions of this theory that gives can be calculated using this type of expansion as before, where these coefficients grow as in factorial, but they are functions of diabolical physical quantities like the moments and the energy of the beams and these large colliders and it turns out that , although it is an asymptotic expansion whose coefficients grow as factorial, the couplings here depend on the scale of the phenomena being measured and for a sufficiently large impulse energy z', these couplings can become, in In principle, you want to buy enough energies and, although this is so, as long as you take P larger and larger, these Alpha coefficients will get smaller, in principle you have predictions that get better and better as you run experiments at higher and higher energies.

So that's one reason and you know there are millions and millions of predictions. They are used all the time in calculation processes for the LHC. They are background, uninteresting things, but the ability to calculate a crown and have those calculations work are extraordinary tests. of the theories, so that's one of the reasons why we believe that this theory exists and its predictions are valid and that these perturbative expansions make sense now because of the way in which a way that has not been used until now to construct the theory correctly in the end turns out to be To be a proof of both the existence and its properties is to make mathematical sense of these deviation expansions, in the same way that you can make mathematical sense of other asymptotic series for mathematical objects if you can control the various singularities that you get when you build Borel transforms. or other things, in fact there is a revived interest in making physical sense of such expansions and thatIt might actually be a different way than people used to define the theory and prove its existence, but the way people used to prove our physics.

From the point of view of the existence of a pet and calculating the pet, which is the same as calculating the spectrum of the theories and masses of elementary particles, is to define the physical observables in terms of a truncated discretized version of the theory and then take a limit, so this is based on path integrals that physicists used all the time to derive beautiful mathematical results that really have no mathematical basis except as limits of finite-dimensional integrals, so in the case of gauge theory it is necessary to replace this action which is generally a path integral and this is going to happen both in quantum mechanics, briefly speaking, the basic entities that are products of fields or products of fields that are based on points can be calculated by integrals , the path integrals call integrals over the basic field variables furnace phase which is just the action of those field variables and then the operators express these functionals of those field variables.

Such path integrals can only really be defined for quantum mechanical systems of a finite number of degrees of freedom and in the case of infinite dimension to obtain a large number of beginnings of a well-defined expression, one first rotates time to imaginary time or our three one two is four three places Mikowski space times Euclidean space and then there is a well-defined procedure for computing these so-called correlation functions in the Clinton space going back to the so-called wipin functions or the objects that can be used to construct a space of Hilbert and the physical observables of a quantum field theory, the reason you do this is that then you have a real measure here and you can Think about evaluating this integral, not Merrick Lee, but first you need to replace this product over functions by an ordinary integral of products over variables at each space-time point in some lattice that approximates four-dimensional space, and to do this you have to introduce some kind of space-time discretization with a lattice separation and then the action will depend on the fields and this network spacing now in the case of Abelian gauge theories or gauge theories in general, the variables are these connections, a connection really tells you how it feels like transport gives you a lot of a point to another from the transformation of the group at one point to that of another and therefore in the lattice definition of gauge theories, the basic variables live in these links and our unitary operators, so one has some kind of network In four dimensions, the field intensity, the curvature tells you what happens if you do parallel transport around a small loop, so the field intensity is defined in a black hat, a small square, and it will be the product of these operators unitary arrays in the group around this little socket and what replaces this action is simply a unitary matrix that describes what happens when you spin around a small cube and this adds to all the cubes, so for each socket you have, in In a certain sense, that of a link is the parallel transport along the link of the meter connection and its P, if you work this out, is approximately something like the exponential of I multiplied by F DX. mu DX nu where is this cube in the new direction my sum over all layer cats take the trace of F a0 the first surviving term here is the sum approximately of all the pockets from a to the fourth F to the four F squared more water terms a to the sixth and this is only if a kit becomes arbitrarily small just like this integral, so this is a discretization of the action and in terms of these plated variables that are products of linked variables and one se integrates into all the links in the network with our measure being bounded and for a finite network a finite number of links, this is certainly a well-defined row, the idea is that as this network expands to fill all the space and becomes finer and finer, this should reach up to Terms that are irrelevant disappeared as powers of a squared compared to this term.

This is like the integral over trace space F, where if these terms rotate, these higher order terms have additional powers of a and easily go to zero, but this Define a continuum theory now. I've gone through all of this, unless you've seen this before you'll never understand it, but it essentially replaces spacetime with a network of points in spacetime that you could put into periodic boundary conditions, it helps numerically. You do this on a torus and then you want to use it as a well-defined measure to calculate operator correlation functions and the only problem is that you have to carefully choose this coupling that appears here as the analogue of the fine structure constant depending also on the spacing of lattice now the notable property of an abelian gauge series is how you have to choose the dependence of the coupling on a at the cut when you take the zero cut, you want to take all of this in the limit where this lattice of points fills the space and a is going to zero the famous Virgin says ultraviolent divergences the interesting dynamical behavior of quantum field theories has to do with the behavior of a if you let the continuum end, let the lattice spacing or cutoff go to zero and Here it is where the big difference occurs between a theory like QED and a theory like UCD, so in QCD there is a definitive prescription that you have to thank when it goes to zero for this limit to exist and that requires that the coupling be defined. in this scale network, a disappears as 1 over log sometimes, some dimension parameter can actually define it as 1 over lambda when a goes to zero, this explodes, this disappearance is like what over log and lambda is an arbitrary parameter of dimensions 1 over length or mass that you introduce to define this limit, what we know is that if you evaluate the correlation function using this measure through perturbation theory expanding around the Gaussian fixed point, whose point is quadratic, then you can calculate perturbatively all these sums and that is known as perturbation theory and in that boundary turbit you can show that if you let G square disappear by coupling that off scale, it disappears logarithmically like this, then term by term in that expansion all the physical observables are finite, furthermore, even though those perturbative expansions are asymptotic, you can always go to small f where this behavior makes G as small as you want, which means you can think that you can rely on an asymptotic expansion and if you are a coupling you don't It is small enough. you just make the network a little smaller and then it gets better and better, so there are strong reasons to believe that this so-called asymptotic freedom, the disappearance of a coupling that governs the scale of interactions at a distance a, disappears when a goes to zero that that is what is needed for this limit to exist that innocence is what is needed the first thing is to prove that the limit exists the limit exists to use this measure of course inserting here some appropriate product of observables formulated in terms of these lattice approximations to field strength Now, if you could show that the limit exists, then you could probably also prove all the properties of the expectation correlation functions.

You know the multi-operator correlation functions you need to satisfy the construction axioms of the quantum filter. The existence of a unique vacuum means that you would have to prove that the theory has symmetry. The Lorentz invariants of the original theory you have broken by putting the theory last. We just all have our discrete subgroup of the Lorentz group notation group. and translation streams which we believe is true because the terms that violate the Lorentz invariants are these terms that have additional powers of lattice spacing and order by order and Britta Bayesian theory does it by saddle point order by order in the saddle point approximation In fact, they disappear as powers in a way, so order by order in this asymptotic expansion it gets better and better as it breaks down and leans or tangles.

Everything works, but that is not proof that the limit exists for physics or approval, but it is not mathematical proof. test and now there have been some advances by Balaban and some other constructive field theorists who went quite far, I think for non-Abelian gauge theory in a finite box, so in a finite box the size of the system is maintained of the torus if what you want to fix is ​​to take the lattice spacing to zero and, from a physical point of view, that makes the job much easier because, in a finite box, you don't have this problem of a continuum, you have discrete states and you never get through a region, you can make the box small enough, so that if the asymptotic expansion gets better and better and you can put downs and control, try to control a limit, but even there no one really doesn't It is a proof of existence, even in a box that I know and the documents are really impossible to read and, in general, workers prevented them from that effort, as far as I know, in the last decade.

Now, that's just the first part that proves that the limit exists, the next part actually proves that there is a massive gap. and there what you have to do is do what people do anyway, which is calculate the mass spectrum of this theory and they do it and one of the reasons why we believe in the theory beyond perturbative asymptotic expansion is that by doing numerical Monte Carlo integration you integrate these integrals and then try to take the limit for a larger and larger lattice, they can calculate the masses and the fine is that as you take this limit, chi-square disappears as 1 over a, you have introduced some parameter with dimensions of mass 1. about the length and that the masses are calculated by you and there is a specific procedure to make it our finite and our numbers depend on the state as the pure number in units of this arbitrary scale that you have introduced to define this limit now you say what the scale is I don't know, let's put an equal to 1 with nothing compared to because this theory has no scale, so when you have a theory in which you have introduced the scale in this way to define the theory, then all masses are numbers multiplied by this scale, that is what we call dimensional transport, we have changed this dimensionless number to a full dimension number and there is no point in saying what the value of this is, it just is what it is, take one of these masses, one of these particles, say the one in the In the real world, we could take the proton and call it mass one and then all the other masses particle 1 particle to the proportions will just be pure numbers and this is what people when calculating compares with the experiment and now, after 30 years or more of doing these calculations.

With enormous effort, we have calculated the spectrum of all the low hadron particles made mostly of quarks which I have not seen, yes, but if in pure QCD, which we cannot compare with the experiment, it would be the masses of particles made of these interacting quanta from the gauge field to one and agree with experiment with an accuracy of one percent also in their various ways of estimating theoretical errors, but there is nothing rigorous about this, no one has ever shown that the limit exists and no one has a rigorous way to control them. errors, however, we have enough, you know, all states that exist in nature appear on the spectrum, there are no states with zero mass, so there is a gap, in fact, there is no miracle e and that is why we believe That the answer should be yes, this limit exists and yes, it is calculated.

On the energy spectrum there is a gap and an interesting spectrum, so unfortunately I didn't have time to discuss this remarkable phenomenon that is the basis of this. If you ask what happens with QED quantum electrodynamics, you discover order by order and perturbation theory. You have to organize this coupling based on the network space so that it gets bigger and bigger, which means that you don't trust what you are doing because the expansion of the disturbance starts to diverge and everything indicates that before reaching the limit continuous. before reaching zero this parameter explodes and you lose all sense of what you are doing, it doesn't mean that this QED theory doesn't describe the real world, we know it does, but something is missing, you need other ingredients, you might need one. or two other ingredients could be part of string theory could be part of a larger unified theory could involve an infinite number for we have no idea by itself it simply means that it is not what we would call a complete UV theory that is extendable to distances arbitrary short, in fact in four dimensions, we believe that the only theories that have a chance of existing in this sense are these very special theories and all of them are contained in the definition of theclay problem, so they are not just any alcoholic.

Field theories describe the real world: the only areas that manage to do all these miracles and therefore deserve testing. Everything we know is true. Actually, thank you. So mathematicians of any question, I'm sure they started talking about them. you won and how mathematically it's a sick theory in some ways, yes, but then of course you wrote G between 2 and 13 or 14 decimal places, right? Would you say this has to do with the connection between mathematics and physics? I guess so. you say that G over 2 is a, is it a number in the sense of mathematics because it is not defined in place 137 or something like that couldn't?

Could you say that? So there is a measure where the particle mass ratio is. we discussed it here, I discussed it here by the way, vacuum QCD, if you are QCD regardless, there are no quarks, only the non-linear interaction fields which in turn are charged, they are very complicated and in the real world there are also quarks and, unfortunately, that introduces other parameters. but if we didn't have those quarks, we just had this theory, there are these numbers that are not going to be simple rational numbers or anything that is known to be mathematically good, but they were, they are numbers, an infinite set of QCD numbers, this QCD without matter. with a dimensionless parameter that looks like a number of freedom you could put in, you end up with a scale, a scale that means nothing because you're a nine in comparison, so all dimensionless measure lasting quantities in the qcdr calculus, so mathematically All of these are interesting.

The numbers are pure numbers and depend on the gauge group G and nothing else on the gauge if the gauge group is a problem and they have an interesting dependence on n, for example, one that we exploit because life gets simpler and bigger, but They are just numbers with a gauge group characteristic, if one had some really powerful mathematical way of describing this dynamics, they could turn out to be very interesting mathematically, certainly, from the point of view of physics, there are very few theories that we have in physics that have such a rich and powerful structure where there is an infinite set of quantities that are just very specific numbers.

I have a naive question: the existence of the mass calf has something to do with the Namibian nature of the caliber group yes, definitely remember that there is only one abelian theory and it does not have a mascot for sure so this has to do there are forms of the theory abelian is trivial there are two trivial to have a space in the vacuum they are just free waves with continuous spectrum the non-abelian theory is interactive theory and there are many ways to understand where the gap comes from and the dynamics that create these linked massive states, but not is that many Abelian theories and all non-Abelian theories must have the same behavior, well, you know that.

It's also ambitious in the sense that you don't know what the real world requires instead of less than four dimensions that make life easier or additional symmetries like supersymmetry that make life easier and also, but it takes the easy way out. We do not ask that this be true on an arbitrary manifold, but we have very good reason to believe that if quantum field theory exists in a flat space with a flat metric, small deformations of the background geometry are exchanged because all the problems occur. . you know in the vicinity at very short distances, as long as you deform the geometry in a smooth way, that again is a popular theorem, but it is surely true, so this would be the starting point now that the applications are quantum theory of fields, which I have been of great interest to mathematicians and I suppose they would like to have a firm basis for some of those applications.

You often know that they don't work in flat space, they work in some curved variety, but as long as the curvature is, you know. smooth, then locally the theory is flat, it won't change, so I imagine that once, if you had methods powerful enough to prove the existence of a flat space, you could test it in an arbitrary smooth well, but you know, like in this definition, you don't really use what you use is that you break the Lorentz in symmetry and by defining the object you recover it in the limit yes, it is difficult, it is much more difficult, especially if if you have something you know that the global structure of space- time is different from